Use a commercial grade gutter sealant and simply re apply where needed in a thorough manner.
Patching gutter leaks.
Then use a power washer or the strongest blast from your.
Remove any screws or connecting hardware at the joint and then disassemble the leaking portion of the downspout.
If it is large use a piece of galvanized metal flashing.
2 clean the joints replace the gaskets.
If the hole is small you can use plastic roofing cement.
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It is easy to fix the leaking rusted seams where gutters connect to each other.
Any issues like this need to be resolved before new sealant can be applied.
Using a small trowel a gutter scoop or your gloved hands clear away as much debris as possible dropping it into a bucket for disposal.
Dry the area with a rag as it must be completely dry for the caulking or roof cement to adhere.
Leak stopper rubber flexx has a great 2 part system with waterproof tape and liquid coating.
Finally thoroughly clean the area of dirt debris shingle granules etc so that the new repair will hold properly.
You may need to remove other gutter or downspout sections near the leaky joint first.
Leaky gutters are easy to repair.